Proof Inviscid Burgers equation is hyperbolic












1












$begingroup$


I'm having problems classifying the inviscid burgers equation into elliptic, parabolic or hyperbolic:




$u_t+uu_x=f(x,t)$




Since the classification for 2nd order partial differential equations of the form:




$a_{11}(x,y)u_{xx}(x,y) + 2a_{12}(x,y)u_{xy}(x,y)+a_{22}(x,y)u_{yy}(x,y) + b_{1}(x,y)u_{x}(x,y) + b_{2}(x,y)u_{y}(x,y) + c(x,y)u(x,y) = d(x,y)$




takes place defining:




$Delta=a_{12}^2-a_{11}a_{22}$




And studying its sign so:




$Delta=0 to$ parabolic equation
$Delta>0 to$ hyperbolic equation
$Delta<0 to$ elliptic equation




I guessed this equation would be parabolic since there aren't any second order derivatives so:




$a_{11}=a_{12}=a_{22}=0$




But checking in the internet it is said to be a hyperbolic equation, and I don't have a clue why it is that way.



Can someone help me?
Thank you very much.






pde partial-derivative







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$endgroup$

















    1












    $begingroup$


    I'm having problems classifying the inviscid burgers equation into elliptic, parabolic or hyperbolic:




    $u_t+uu_x=f(x,t)$




    Since the classification for 2nd order partial differential equations of the form:




    $a_{11}(x,y)u_{xx}(x,y) + 2a_{12}(x,y)u_{xy}(x,y)+a_{22}(x,y)u_{yy}(x,y) + b_{1}(x,y)u_{x}(x,y) + b_{2}(x,y)u_{y}(x,y) + c(x,y)u(x,y) = d(x,y)$




    takes place defining:




    $Delta=a_{12}^2-a_{11}a_{22}$




    And studying its sign so:




    $Delta=0 to$ parabolic equation
    $Delta>0 to$ hyperbolic equation
    $Delta<0 to$ elliptic equation




    I guessed this equation would be parabolic since there aren't any second order derivatives so:




    $a_{11}=a_{12}=a_{22}=0$




    But checking in the internet it is said to be a hyperbolic equation, and I don't have a clue why it is that way.



    Can someone help me?
    Thank you very much.






    pde partial-derivative







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      I'm having problems classifying the inviscid burgers equation into elliptic, parabolic or hyperbolic:




      $u_t+uu_x=f(x,t)$




      Since the classification for 2nd order partial differential equations of the form:




      $a_{11}(x,y)u_{xx}(x,y) + 2a_{12}(x,y)u_{xy}(x,y)+a_{22}(x,y)u_{yy}(x,y) + b_{1}(x,y)u_{x}(x,y) + b_{2}(x,y)u_{y}(x,y) + c(x,y)u(x,y) = d(x,y)$




      takes place defining:




      $Delta=a_{12}^2-a_{11}a_{22}$




      And studying its sign so:




      $Delta=0 to$ parabolic equation
      $Delta>0 to$ hyperbolic equation
      $Delta<0 to$ elliptic equation




      I guessed this equation would be parabolic since there aren't any second order derivatives so:




      $a_{11}=a_{12}=a_{22}=0$




      But checking in the internet it is said to be a hyperbolic equation, and I don't have a clue why it is that way.



      Can someone help me?
      Thank you very much.






      pde partial-derivative







      share|cite|improve this question









      $endgroup$




      I'm having problems classifying the inviscid burgers equation into elliptic, parabolic or hyperbolic:




      $u_t+uu_x=f(x,t)$




      Since the classification for 2nd order partial differential equations of the form:




      $a_{11}(x,y)u_{xx}(x,y) + 2a_{12}(x,y)u_{xy}(x,y)+a_{22}(x,y)u_{yy}(x,y) + b_{1}(x,y)u_{x}(x,y) + b_{2}(x,y)u_{y}(x,y) + c(x,y)u(x,y) = d(x,y)$




      takes place defining:




      $Delta=a_{12}^2-a_{11}a_{22}$




      And studying its sign so:




      $Delta=0 to$ parabolic equation
      $Delta>0 to$ hyperbolic equation
      $Delta<0 to$ elliptic equation




      I guessed this equation would be parabolic since there aren't any second order derivatives so:




      $a_{11}=a_{12}=a_{22}=0$




      But checking in the internet it is said to be a hyperbolic equation, and I don't have a clue why it is that way.



      Can someone help me?
      Thank you very much.






      pde partial-derivative




      pde partial-derivative






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      asked Jan 31 at 18:06









      andreanapoliandreanapoli

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          $begingroup$

          Your equation is of the first order. You cannot use the second order condition to check whether it is hyperbolic/parabolic/elliptic.



          You can read about the classification of first order equations/systems in this pdf:



          https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=2ahUKEwjGnO_I4JngAhWTHDQIHWy2DE4QFjAEegQIBRAC&url=http%3A%2F%2Fpeople.3sr-grenoble.fr%2Fusers%2Fbloret%2Fenseee%2Fmaths%2Fenseee-maths-IBVPs-3.pdf&usg=AOvVaw3iWOWyyz7PyfihM3IzGIxH






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            $begingroup$

            Your equation is of the first order. You cannot use the second order condition to check whether it is hyperbolic/parabolic/elliptic.



            You can read about the classification of first order equations/systems in this pdf:



            https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=2ahUKEwjGnO_I4JngAhWTHDQIHWy2DE4QFjAEegQIBRAC&url=http%3A%2F%2Fpeople.3sr-grenoble.fr%2Fusers%2Fbloret%2Fenseee%2Fmaths%2Fenseee-maths-IBVPs-3.pdf&usg=AOvVaw3iWOWyyz7PyfihM3IzGIxH






            share|cite|improve this answer









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              0












              $begingroup$

              Your equation is of the first order. You cannot use the second order condition to check whether it is hyperbolic/parabolic/elliptic.



              You can read about the classification of first order equations/systems in this pdf:



              https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=2ahUKEwjGnO_I4JngAhWTHDQIHWy2DE4QFjAEegQIBRAC&url=http%3A%2F%2Fpeople.3sr-grenoble.fr%2Fusers%2Fbloret%2Fenseee%2Fmaths%2Fenseee-maths-IBVPs-3.pdf&usg=AOvVaw3iWOWyyz7PyfihM3IzGIxH






              share|cite|improve this answer









              $endgroup$
















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                0








                0





                $begingroup$

                Your equation is of the first order. You cannot use the second order condition to check whether it is hyperbolic/parabolic/elliptic.



                You can read about the classification of first order equations/systems in this pdf:



                https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=2ahUKEwjGnO_I4JngAhWTHDQIHWy2DE4QFjAEegQIBRAC&url=http%3A%2F%2Fpeople.3sr-grenoble.fr%2Fusers%2Fbloret%2Fenseee%2Fmaths%2Fenseee-maths-IBVPs-3.pdf&usg=AOvVaw3iWOWyyz7PyfihM3IzGIxH






                share|cite|improve this answer









                $endgroup$



                Your equation is of the first order. You cannot use the second order condition to check whether it is hyperbolic/parabolic/elliptic.



                You can read about the classification of first order equations/systems in this pdf:



                https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&ved=2ahUKEwjGnO_I4JngAhWTHDQIHWy2DE4QFjAEegQIBRAC&url=http%3A%2F%2Fpeople.3sr-grenoble.fr%2Fusers%2Fbloret%2Fenseee%2Fmaths%2Fenseee-maths-IBVPs-3.pdf&usg=AOvVaw3iWOWyyz7PyfihM3IzGIxH







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Feb 1 at 6:32









                GReyesGReyes

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